"By oneself" versus "on one's own"

Question

Is there any difference between these three pairs? If there is, is it the same difference?

I did it by myself.

I did it on my own.

 

He lives by himself.

He lives on his own.

 

We're by ourselves.

We're on our own.

Answer 1

In looking into questions of synonymy (or its absence) it is important to keep in mind that individual words and phrases themselves do not have atomic meaning: they have not only ranges of meanings, and different meanings in different contexts, but also different ranges of meanings in different contexts. So even in such distinct contexts as you provide you may discover instances where the two phrases are essentially synonymous, instances where only one or the other phrases is usable, and instances where both may be used but they have different meanings. There is not necessarily a 'core meaning' to any word/phrase, or 'core difference' between two words/phrases, which translates predictably from one context to the next.

Let's look at the phrases 1) by Xself and 2) on X own in three different contexts: a) with do, b) with live, and c) with be.

1a: I did it by myself.
2a: I did it on my own.

• Both by myself and on my own may mean ‘without assistance from anybody else’. But by myself may also mean ‘when I was alone’; and on my own may also mean ‘on my own initiative, without being prompted or ordered to do so’.

1b: He lives by himself.
2b: He lives on his own.

• Both by himself and on his own here may mean simply ‘without any cohabitant’. But by himself may be used with an implication of isolation or loneliness, while on his own may be used with an implication of independence and self-sufficiency.

1c: We're by ourselves.
2c: We're on our own.

• Both by ourselves and on our own be used to mean ‘without assistance or support’, ‘left to our own devices’; but on our own may also express a sense of opportunity to exercise initiative. And by ourselves is more often used to signify privacy—“Alone at last!”—and consequent freedom to pursue more intimate or surreptitious activities.

Answer 2

I agree with @FumbleFingers on the 'nobody is here except us' / 'nobody will help us' difference. Further more:

1: I did it by myself can convey a positive meaning (you're proud of yourself)

while

2: I did it on my own tends to convey a negative one (nobody was here to help me so I had to do it... by myself)

In your 4rd example:

He lives on his own

If I had to imagine a situation, I would say that he is a young adult and life sad circumstances (like the death of his parents) made him rent a house. He had no source of care.

While the 3rd example

He lives by himself

Makes me imagine he's an independent young adult that took the opportunity to live alone.

Answer 3

As far as I could find, both mean pretty much the same thing and can be used interchangeably. But there's a small difference:

1. by myself is mostly used to mean without help from someone else.
2. on my own is mostly used for a situation (but could be used for the without help meaning, either).